Comparison of Deep Learning and Deterministic Algorithms for Control Modeling

Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm

Surrogate modeling of parameterized multi-dimensional premixed combustion with physics-informed neural networks for rapid exploration of design space

Discretization-based methods like the finite element method have proven to be effective for solving the Helmholtz equation in computational acoustics. However, it is very challenging to incorporate measured data into the model or infer model input parameters based on observed response data. Machine learning approaches have shown promising potential in data-driven modeling. In practical applications, purely supervised approaches suffer from poor generalization and physical interpretability. Physics-informed neural networks (PINNs) incorporate prior knowledge of the underlying partial differential equation by including the residual into the loss function of an artificial neural network. Training the neural network minimizes the residual of both the differential equation and the boundary conditions and learns a solution that satisfies the corresponding boundary value problem. In this contribution, PINNs are applied to solve the Helmholtz equation within a two-dimensional acoustic duct and mixed boundary conditions are considered. The results show that PINNs are able to solve the Helmholtz equation very accurately and provide a promising data-driven method for physics-based surrogate modeling.

The accurate modeling of the Earth gravity field and geoid is critical for geodesy, yet traditional methods face limitations in handling the growing complexity and heterogeneity of modern geodetic data. To address these challenges, this study proposes a physics-informed neural network (PINN) framework for high-precision geoid modeling. The PINN employs convolutional neural networks (CNNs) to extract multi-scale features from terrestrial and airborne gravity data, which are then processed by a multilayer perceptron (MLP) to establish an accurate mapping between these features and the disturbing potential. Physical constraints, including Laplace’s equation and differential equations governing gravity anomaly and gravity disturbance, are embedded into the loss function to enhance both accuracy and interpretability. The proposed method is applied to the Colorado 1 cm geoid experiment. Compared to GNSS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17), the PINN-derived geoid model achieves a standard deviation (STD) of 2.1 cm. This represents a 12.5%–27.6% improvement over traditional methods and purely data-driven networks (DDNs). The PINN exhibits strong generalization under sparse data conditions, achieving 28.5% higher accuracy than the DDN with only 500 samples. Furthermore, analysis of geoid slopes and physical constraint contributions demonstrates that PINN’s dual physical constraints effectively balance global characteristics and localized fidelity of the geoid. This study establishes the PINN as a robust, physically interpretable machine learning approach for geoid modeling, outperforming classical methods and offering a promising pathway for gravity field estimation in regions with sparse or heterogeneous data. By bridging purely data-driven machine learning with fundamental geodetic principles, this work paves the way for future advancements in physics-informed machine learning-based geodetic modeling.

Summary Classical reservoir engineering studies require building geological models and solving complex fluid flow transport equations that require high-quality data, numerous computational resources, time and workflows. For large and mature fields data-driven models can be used to get faster answer and to perform production analysis more efficiently. Capacitance Resistive Models (CRM) are a class of methods based on material balance that can be used to estimate production wells liquid rates as a function of injected water and Bottom Hole Pressure (BHP) variations. CRM methods quantify the connectivity between producers and injectors using only dynamic data. An important drawback of CRM is that they can suffer from parameter identification problems. Moreover, the analytical solution can be only obtained in specific conditions: linear variations of BHP and fixed injection rate between two consecutive time steps. In this work we present a new approach combining CRM material balance equations with neural networks in order to obtain more robust and reliable estimation of the CRM parameters (i.e. well connectivity, productivity indices and time constants). This proposal is also interesting since it is not based on any assumption on BHP and injection rates. To this end, we use a recent approach called Physics Informed Neural Networks (PINNs). In this approach neural networks are trained on observed data with additional physics constraints traduced in appropriate loss functions. The parameters of this physical equation are evaluated at the same time as the neural network weights. The introduction of PINNs in our approach raised after testing classical machine learning (ML) models (SVMs, Random Forests …) and deep learning models (MLP, LSTM, RNNs…). Indeed, such models can perform well in some specific cases but usually struggle to produce robust results (i.e. forecasting) in the long term. Unfortunately, such systems do not natively integrate physics constraints. Our aim is to impose physic constraints in neural networks. Thus, we may obtain more stable and reliable results. On the same time, we should be able to account for more behaviors that are not explained by simplified physic equations such as material balance. We performed a full comparison between our approach using PINNs, other standard ML and DL approaches and a given framework of CRMs on two data-sets: a simple but realistic model build using a commercial reservoir simulator, and a real data set. We show that our approach gives more robust results (in terms of MSE) while not suffering from parameter identification issue.

A steady-state nonlinear feedforward controller (FFC) for measurable disturbances is designed for a continuous bioreactor, which is represented by Hammerstein type nonlinear model wherein the nonlinearity is a polynomial with input multiplicities. The manipulated variable is the feed substrate concentration (Sf) and the disturbance variable is the dilution rate (D). The productivity (Q=DP) is considered as the controlled variable. The desired value of Q=3.73 gives two values of feed substrate concentration. The nonlinearity in the gain is considered for relating output to the manipulated variable and separately for the relation between output to disturbance variable. The FFC is also designed for the overall linearized system. The performance of the FFC is evaluated on the nonlinear differential equation model. The FFC is also designed for the model based on a single nonlinear steady-state equation containing both D and Sf. This nonlinear FFC gives the best performance. The nonlinear FFC is also designed by using only linear gain for the disturbance and nonlinear gain for the manipulated variable. Similarly, nonlinear FFC is also designed by using linear gain for the manipulated variable and the nonlinear gain for the disturbance variable. The performances of these FFC schemes are compared.

A novel method based on a physics informed data driven model is developed to design an origami composite crash tube. The structure consists of two axially stacked basic components, called modules. Each module presents lower and upper square sections with an octagonal section in the middle. The parameters of the octagonal cross-section and the height of each module are optimized to maximize the energy absorption of the tube when subjected to an axial impact. In contrast to standard surrogate modelling techniques, whose accuracy only depends on the amount of available data, a Physics-informed Neural Network (PINN) scheme is adopted to correlate the crushing response of the single modules to that of the whole origami tube, constraining the data driven method to physically consistent predictions. The PINN is first trained on the results obtained with an experimentally validated Finite Element model and then used to optimize the structure. Results show that the PINN can accurately predict the crushing response of the origami tube, while consistently reducing the computational effort required to explore the whole design domain. Also, the comparison with a standard Feed Forward Neural Network (FFNN) shows that the PINN scheme leads to more accurate results.

The nonlinear telegraph equation appears in a variety of engineering and science problems. This paper presents a deep learning algorithm termed multi-term physics-informed neural networks to resolve initial boundary value problems of 2D and 3D hyperbolic nonlinear telegraph equations. The standard physics-informed neural networks include the gradients of residuals of the problem, initial conditions, and boundary conditions directly into the training process in addition to the standard PINNs loss function terms. Using multiple densely connected neural networks, termed feed-forward deep neural networks, the proposed scheme has been trained to minimize the total loss results from the multi-term loss function. Three computational examples are provided to demonstrate the efficacy and applications of our suggested method. For each illustrative example considered, we compared the proposed scheme with the standard physics-informed neural networks based on the network model train and test losses and other well-known performance measure error analysis including each problems with long-time domain. The results demonstrate that the proposed method outperforms the conventional deep learning approach physics-informed neural networks in predicting the solutions of both initial boundary value problems of telegraph equations. Therefore, the proposed computational approach can offer precise and stable solutions for long-time domain PDEs and is efficient in handling challenging nonlinear physical problems with a variety of boundary conditions.

The classical step-by-step integration method is commonly used for solving structural dynamic responses, but its performance is highly sensitive to the choice of time step. A large time step can significantly reduce accuracy and stability, while a very small time step greatly diminishes computational efficiency. Data-driven modeling and analysis methods are particularly favored for their potential to replace traditional structural analysis methods. However, traditional data-driven machine learning approaches require substantial amounts of training data to predict structural dynamic responses. Moreover, these methods often function as black boxes, lacking physical interpretability, and acquiring the necessary data can be challenging in practice. To address these limitations, this paper employs a physics-informed neural network (PINN) method for structural dynamic response analysis. The PINN approach incorporates the governing equations of structural dynamics into the neural network’s loss function as physical constraints, endowing the model with physical interpretability. The feasibility and effectiveness of this method are demonstrated through case studies on structural dynamic responses under different excitation scenarios. Numerical results show that the PINN method can accurately analyze structural dynamic responses without relying on system response data, a key advantage over traditional data-driven models. Moreover, unlike the classical step-by-step integration method, the PINN approach operates as a spatial domain optimization process and is independent of time step. Given the critical role of hyperparameters in the performance of the PINN model, this paper systematically investigates the influence of various hyperparameter settings on accuracy and efficiency, uncovering patterns of their impact on structural response analysis. Replication code for this paper will be available on GitHub:https://github.com/duke-iem/AIStructDynSolve.

A nonlocal physics-informed deep learning framework using the peridynamic differential operator

Feedforward (FF) control uses a priori knowledge about a given system and its disturbances to influence the system’s behavior in a pre-defined way. However, unlike feedback (FB) control, it does not adjust the control signal (or manipulated variable) in response to how the system actually reacts. In other words, FF control is proactive while FB control is reactive. Model-inversion-based FF (MIBFF) control is a specific type of FF control where the control variable is determined by inverting a model of the system to be controlled. It is very popular for its use in output tracking control problems, where the goal is to force the system’s output to follow a desired trajectory. Output tracking is critical in several different applications, e.g., in manufacturing, robotics, automotive, aerospace, and is typically achieved using a combination of FF and FB control. As motivating examples, consider MIBFF in ultra-precise wafer scanners used in photolithography (see Sidebar S1) and in desktop 3D printing (see Sidebar S2). As these motivating examples illustrate, non-minimum phase (NMP) zeros often arise in practical engineering applications due to non-collocated sensors and actuators, fast sampling, etc. MIBFF for systems with NMP zeros is arguably the most important (and certainly the most researched) issue related to MIBFF.

This paper illustrates the synthesis of nonlinear feedforward controllers for chemical reactors. In most of the theoretical development and application of feedforward control only linear systems have been considered. There are, however, no inherent linear limitations in feedforward control. Since chemical reactors are usually nonlinear, the effectiveness of control should be improved by including nonlinearities in the design of feedforward controllers. This is particularly true for batch reactors because of the large changes in variables during a batch cycle. Continuous stirred‐tank reactors are studied with single and consecutive reactions of first and higher order. Effectiveness of linear and nonlinear feedforward controllers is compared for disturbances of various magnitude and direction. Feedforward control of batch and tubular reactors is also discussed.

The cerebellum has been hypothesized to form a crucial part of the speech motor control network. Evidence for this comes from patients with cerebellar damage, who exhibit a variety of speech deficits, as well as imaging studies showing cerebellar activation during speech production in healthy individuals. To date, the precise role of the cerebellum in speech motor control remains unclear, as it has been implicated in both anticipatory (feedforward) and reactive (feedback) control. Here, we assess both anticipatory and reactive aspects of speech motor control, comparing the performance of patients with cerebellar degeneration and matched controls. Experiment 1 tested feedforward control by examining speech adaptation across trials in response to a consistent perturbation of auditory feedback. Experiment 2 tested feedback control, examining online corrections in response to inconsistent perturbations of auditory feedback. Both male and female patients and controls were tested. The patients were impaired in adapting their feedforward control system relative to controls, exhibiting an attenuated anticipatory response to the perturbation. In contrast, the patients produced even larger compensatory responses than controls, suggesting an increased reliance on sensory feedback to guide speech articulation in this population. Together, these results suggest that the cerebellum is crucial for maintaining accurate feedforward control of speech, but relatively uninvolved in feedback control.SIGNIFICANCE STATEMENT Speech motor control is a complex activity that is thought to rely on both predictive, feedforward control as well as reactive, feedback control. While the cerebellum has been shown to be part of the speech motor control network, its functional contribution to feedback and feedforward control remains controversial. Here, we use real-time auditory perturbations of speech to show that patients with cerebellar degeneration are impaired in adapting feedforward control of speech but retain the ability to make online feedback corrections; indeed, the patients show an increased sensitivity to feedback. These results indicate that the cerebellum forms a crucial part of the feedforward control system for speech but is not essential for online, feedback control.

To estimate the contributions of feedforward vs. feedback control systems in speech articulation, we analyzed the correspondence between initial and final kinematics in unperturbed tongue and jaw movements for consonant-vowel (CV) and vowel-consonant (VC) syllables. If movement extents and endpoints are highly predictable from early kinematic information, then the movements were most likely completed without substantial online corrections (feedforward control); if the correspondence between early kinematics and final amplitude or position is low, online adjustments may have altered the planned trajectory (feedback control) (Messier and Kalaska, 1999). Five adult speakers produced CV and VC syllables with high, mid, or low vowels while movements of the tongue and jaw were tracked electromagnetically. The correspondence between the kinematic parameters peak acceleration or peak velocity and movement extent as well as between the articulators' spatial coordinates at those kinematic landmarks and movement endpoint was examined both for movements across different target distances (i.e., across vowel height) and within target distances (i.e., within vowel height). Taken together, results suggest that jaw and tongue movements for these CV and VC syllables are mostly under feedforward control but with feedback-based contributions. One type of feedback-driven compensatory adjustment appears to regulate movement duration based on variation in peak acceleration. Results from a statistical model based on multiple regression are presented to illustrate how the relative strength of these feedback contributions can be estimated.

Self-induced stochastic resonance (SISR) is the emergence of coherent oscillations in slow-fast excitable systems driven solely by noise, without external periodic forcing or proximity to a bifurcation. This work presents a physics-informed machine learning framework for modeling and predicting SISR in the stochastic FitzHugh–Nagumo neuron. We embed the governing stochastic differential equations and SISR-asymptotic timescale-matching constraints directly into a Physics-Informed Neural Network (PINN) based on a Noise-Augmented State Predictor architecture. The composite loss integrates data fidelity, dynamical residuals, and barrier-based physical constraints derived from Kramers’ escape theory. The trained PINN accurately predicts the dependence of spike-train coherence on noise intensity, excitability, and timescale separation, matching results from direct stochastic simulations with substantial improvements in accuracy and generalization compared with purely data-driven methods, requiring significantly less computation. The framework provides a data-efficient and interpretable surrogate model for simulating and analyzing noise-induced coherence in multiscale stochastic systems. • A physics-informed neural network that learns stochastic dynamics directly. • Noise-Augmented State Predictor (NASP) for data-efficient multiscale modeling. • Predicts noise-driven oscillations and coherence from short trajectory data. • Shown on self-induced stochastic resonance in FitzHugh–Nagumo neuron model.